In this section, we review the basics of Lagrangian mechanics and the derivation of adaptive time-step variational integrators. Drawing on the work of Marsden et al. [1]}, [2]}, [3]}, we first derive equations of motion in continuous-time from the variational principle and then derive the corresponding variational integrators by considering the discretized variational principle in the discrete-time domain. To this end, we first derive continuous-time Euler-Lagrange equations of motion from Hamilton's principle. After deriving equations of motion, we use concepts of discrete mechanics developed in [3]} to derive discrete Euler-Lagrange equations and then write them in the time-marching form to obtain adaptive time-step variational integrators for Lagrangian systems.
